Chapter 3

The polynomial \(P(x)=3(x-5)^{3}(x-3)(x+2)\) has degree ____. It has zeros \(5,3,\) and ____. The zero 5 has multiplicity ____, and the zero 3 has multiplicity ____.

The imaginary number \(i\) has the property that \(i^{2}=\) _________________.

If the polynomial function $$P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ has integer coefficients, then the only numbers that could possibly be rational zeros of \(P\) are all of the form \(\frac{p}{q},\) where \(p\) is a factor of _____ and \(q\) is a factor of _____. The possible rational zeros of \(P(x)=6 x^{3}+5 x^{2}-19 x-10\) are _____.

If we divide the polynomial \(P\) by the factor \(x-c\) and we obtain the equation \(P(x)=(x-c) Q(x)+R(x),\) then we say that \(x-c\) is the divisor, \(Q(x)\) is the ______, and \(R(x)\) is the ______.

(a) If \(a\) is a zero of the polynomial \(P,\) then ____must be a factor of \(P(x)\). If \(a\) is a zero of multiplicity \(m\) of the polynomial \(P,\) then ____ must be a factor of \(P(x)\) when we factor \(P\) completely.

If the rational function \(y=r(x)\) has the horizontal asymptote $$y=2, \text { then } y \rightarrow$$ __________ $$\text { as } x \rightarrow \pm \infty$$.

For the complex number \(3+4 i\) ____________the real part is and the imaginary part is __________.

Using Descartes' Rule of Signs, we can tell that the polynomial \(P(x)=x^{5}-3 x^{4}+2 x^{3}-x^{2}+8 x-8\) has _____, _____, or _____ positive real zeros and _____ negative real zeros.

(a) If we divide the polynomial \(P(x)\) by the factor \(x-c\) and we obtain a remainder of 0 , then we know that \(c\) is a _____ of \(P.\) (b) If we divide the polynomial \(P(x)\) by the factor \(x-c\) and we obtain a remainder of \(k,\) then we know that \(P(c)=\) _____.

The quadratic function \(f(x)=a(x-h)^{2}+k\) is in standard form. (a) The graph of \(f\) is a parabola with vertex ( _______ , ________) (b) If \(a>0,\) the graph of \(f\) opens \(.\) In this case \(f(h)=k\) is the _________ value of \(f\). (c) If \(a<0,\) the graph of \(f\) opens __________. In this case \(f(h)=k\) is the ________ value of \(f\).

Every polynomial has one of the following behaviors: (i) \(y \rightarrow \infty\) as \(x \rightarrow \infty\) and \(y \rightarrow \infty\) as \(x \rightarrow-\infty\) (ii) \(y \rightarrow \infty\) as \(x \rightarrow \infty\) and \(y \rightarrow-\infty\) as \(x \rightarrow-\infty\) (iii) \(y \rightarrow-\infty\) as \(x \rightarrow \infty\) and \(y \rightarrow \infty\) as \(x \rightarrow-\infty\) (iv) \(y \rightarrow-\infty\) as \(x \rightarrow \infty\) and \(y \rightarrow-\infty\) as \(x \rightarrow-\infty\) For each polynomial, choose the appropriate description of its end behavior from the list above. (a) \(y=x^{3}-8 x^{2}+2 x-15\); end behavior _____. (b) \(y=-2 x^{4}+12 x+100 ;\) end behavior ______.

The following questions are about the rational function $$ r(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)} $$ The function \(r\) has \(x\) -intercepts __________ and __________.

A polynomial of degree \(n \geq 1\) has exactly ____ zeros if a zero of multiplicity \(m\) is counted \(m\) times.

(a) The complex conjugate of \(3+4 i\) is \(\overline{3+4 i}=\) _______________ . (b)(3+4 i)(3+4 i)= _________________.

True or false? If \(c\) is a real zero of the polynomial \(P,\) then all the other zeros of \(P\) are zeros of \(P(x) /(x-c)\)

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x).\) $$P(x)=3 x^{2}+5 x-4, \quad D(x)=x+3$$

The graph of \(f(x)=2(x-3)^{2}+5\) is a parabola that opens _________ with its vertex at (_______ _____) and \(f(3)=\) ________ is the (minimum/maximum) _____________ value of \(f\).

If \(c\) is a zero of the polynomial \(P,\) which of the following statements must be true? (a) \(P(c)=0\) (b) \(P(0)=c\) (c) \(x-c\) is a factor of \(P(x)\) (d) \(c\) is the \(y\) -intercept of the graph of \(P\)

The following questions are about the rational function $$ r(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)} $$ The function \(r\) has \(y\) -intercept __________.

If the polynomial function \(P\) has real coefficients and if \(a+b i\) is a zero of \(P,\) then ____ is also a zero of \(P\).

If \(3+4 i\) is a solution of a quadratic equation with real coefficients, then ______________is also a solution of the equation.

True or false? If \(a\) is an upper bound for the real zeros of the polynomial \(P,\) then \(-a\) is necessarily a lower bound for the real zeros of \(P\).

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x).\) $$P(x)=x^{3}+4 x^{2}-6 x+1, \quad D(x)=x-1$$

The graph of \(f(x)=-2(x-3)^{2}+5\) is a parabola that opens _________ , with its vertex at (_______, ______) and \(f(3)=\) ________ is the (minimum/maximum)_________ value of \(f\)

Which of the following statements couldn't possibly be true about the polynomial function \(P ?\) (a) \(P\) has degree \(3,\) two local maxima, and two local minima. (b) \(P\) has degree 3 and no local maxima or minima. (c) \(P\) has degree \(4,\) one local maximum, and no local minimal

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{4}+4 x^{2}$$

The following questions are about the rational function $$ r(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)} $$ The function \(r\) has vertical asymptotes \(x=\) __________ and $$x=$$ __________.

Find the real and imaginary parts of the complex number. $$5-7 i$$

List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). $$P(x)=x^{3}-4 x^{2}+3$$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x).\) $$P(x)=2 x^{3}-3 x^{2}-2 x, \quad D(x)=2 x-3$$

The graph of a quadratic function \(f\) is given. (a) Find the coordinates of the vertex. (b) Find the maximum or minimum value of \(f\). (c) Find the domain and range of \(f\). $$f(x)=-x^{2}+6 x-5$$ (GRAPH CAN'T COPY)

The following questions are about the rational function $$ r(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)} $$ The function \(r\) has horizontal asymptote \(y=\) __________.

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{5}+9 x^{3}$$

Find the real and imaginary parts of the complex number. $$-6+4 i$$

List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). $$Q(x)=x^{4}-3 x^{3}-6 x+8$$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x).\) $$P(x)=4 x^{3}+7 x+9, \quad D(x)=2 x+1$$

The graph of a quadratic function \(f\) is given. (a) Find the coordinates of the vertex. (b) Find the maximum or minimum value of \(f\). (c) Find the domain and range of \(f\). $$f(x)=-\frac{1}{2} x^{2}-2 x+6$$ (GRAPH CAN'T COPY)

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{3}-2 x^{2}+2 x$$

Find the real and imaginary parts of the complex number. $$\frac{-2-5 i}{3}$$

List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). $$R(x)=2 x^{5}+3 x^{3}+4 x^{2}-8$$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x).\) $$P(x)=x^{4}-x^{3}+4 x+2, \quad D(x)=x^{2}+3$$

The graph of a quadratic function \(f\) is given. (a) Find the coordinates of the vertex. (b) Find the maximum or minimum value of \(f\). (c) Find the domain and range of \(f\). $$f(x)=2 x^{2}-4 x-1$$ (GRAPH CAN'T COPY)

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{3}+x^{2}+x$$

Find the real and imaginary parts of the complex number. $$\frac{4+7 i}{2}$$

List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). $$S(x)=6 x^{4}-x^{2}+2 x+12$$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x).\) $$P(x)=2 x^{5}+4 x^{4}-4 x^{3}-x-3, \quad D(x)=x^{2}-2$$

The graph of a quadratic function \(f\) is given. (a) Find the coordinates of the vertex. (b) Find the maximum or minimum value of \(f\). (c) Find the domain and range of \(f\). $$f(x)=3 x^{2}+6 x-1$$ (GRAPH CAN'T COPY)

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{4}+2 x^{2}+1$$

Find the real and imaginary parts of the complex number. $$3$$

List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). $$T(x)=4 x^{4}-2 x^{2}-7$$